A Short Problem that Leads to?

In $\triangle ABC$, $\overline{BC}=a,\overline{CA}=b,\overline{AB}=c$

$P$ is a point in $\triangle ABC$ . Use $a,b,c$ to show the minimum value of $\overline{PA}+\overline{PB}+\overline{PC}$

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

P is actually the Fermat Point of the triangle!!! So, after a bit of calculations, we get the final answer as shown in the link below............( I'm sorry since I don't know LATEX I have just input the answer in Desmos.......)

https://www.desmos.com/calculator/liejprtfic

Aaghaz Mahajan - 2 years, 8 months ago

Thank you! I think you mean $\sqrt{\frac{a^2+b^2+c^2+\sqrt{6((ab)^2+(bc)^2+(ca)^2)-3(a^4+b^4+c^4)}}2}$ when none of the interior angles is bigger than $120^\circ$

If $A$ is the area of the triangle, then it equals $\sqrt{\frac{a^2+b^2+c^2+4\sqrt3A}2}$

X X - 2 years, 8 months ago

Yup.....that is what I have sent in the link!!!! :)

Aaghaz Mahajan - 2 years, 8 months ago

@Aaghaz Mahajan – Thanks!

X X - 2 years, 8 months ago

@X X – @X X No problemmo........anyways, I was thinking, a good follow up to the question would be..........What are the conditions on a b and c such that the given expression is an integer!!! For instance, a triangle with side lengths 3,5 and 7 satisfy the aforementioned property!!!!

Aaghaz Mahajan - 2 years, 8 months ago

@Aaghaz Mahajan – Interesting problem! Let's think about it.

X X - 2 years, 8 months ago

@Aaghaz Mahajan – I looked for Fermat Point and found out also if one of the triangle's angle( $\angle ABC$ ) is bigger than 120 degrees, then the point is on $B$ .

The triangle with side length 3,5,7 has one angle equal to 120 degrees, so this is probably the reason that it is an integer.

(So a triangle with integer lengths, and an angle not smaller than 120 degrees will satisfiy the aforementioned property. But if there is no angle bigger than 120 degrees?)

X X - 2 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$